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Absoluteness, Potential Scott Sentences, and Stability in Model Theory

模型理论中的绝对性、潜在斯科特句子和稳定性

基本信息

批准号：
1855789
负责人：
Michael Chris Laskowski
金额：
$ 21万
依托单位：
University of Maryland, College Park
依托单位国家：
美国
项目类别：
Continuing Grant
财政年份：
2019
资助国家：
美国
起止时间：
2019-06-01 至 2022-12-31
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=1855789&HistoricalAwards=false
关键词：
Absoluteness Potential Scott Sentences Stability

项目摘要

The proposed research is in model theory, which is a branch of mathematical logic. Much of model theory concerns the ways in which a theory, which is simply a set of sentences in a formal language controls its class of models. For many years, the PI has concentrated on mechanisms by which a theory can either admit or forbid certain combinatorial configurations in its models, and the proposed research is to continue these investigations in several disparate contexts. In some cases, this investigation melds well with computational learning theory. As one example, if a theory forbids the independence property, then all of the concepts i.e., definable sets, arising in the models of the theory are PAC learnable.In more detail, potential canonical Scott sentences have proved to be a useful tool in determining the Borel complexity of invariant classes of countable structures and we intend to streamline these methods by exploring thickness and groundedness of classes of models. It is hoped that the Borel complexity of every mutually algebraic theory can be computed. Theories with non-maximal uncountable spectrum are classifiable. Recent technical results about the existence of prime models make it tractable to settle Vaught's conjecture for classifiable theories and possibly for superstable theories as well. In first order logic, aleph1-categoricity of a theory is an absolute notion, as can be seen by the Baldwin-Lachlan characterization of aleph1-categoricity. The PI proposes to determine whether a similar characterization can be found for aleph1-categoricity of sentences of L(omega1, omega), or equivalently for classes of atomic models. Specifically, the proposed research is to determine whether aleph1-categoricity is absolute for classes of atomic models. The PI also proposes to use previous results about mutually algebraic structures to study expansions of weakly minimal structures that preserve stability.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.

模型论是数理逻辑的一个分支。模型论的大部分内容都涉及理论控制模型类的方式，理论只是形式语言中的一组句子。多年来，PI一直专注于一个理论可以在其模型中允许或禁止某些组合配置的机制，而拟议的研究是在几个不同的背景下继续这些调查。在某些情况下，这项研究与计算学习理论融合得很好。作为一个例子，如果一个理论禁止独立属性，那么所有的概念，即，可定义的集合，在理论的模型中产生的PAC learnable.In更详细地说，潜在的规范斯科特句子已被证明是一个有用的工具，在确定可数结构的不变类的博雷尔复杂性，我们打算精简这些方法，探索厚度和crowdedness类的模型。人们希望每一个互代数理论的Borel复杂度都能被计算出来。具有非极大不可数谱的理论是可分类的。最近关于素模型存在性的技术结果使得解决Vaught猜想对于可分类理论和可能对于超稳定理论来说都是容易的。在一阶逻辑中，理论的aleph 1-范畴性是一个绝对的概念，这可以从Baldwin-Lachlan对aleph 1-范畴性的刻画中看出。 PI建议确定是否可以找到类似的表征为aleph 1-范畴的句子L（ω 1，ω），或等价的类原子模型。具体而言，拟议的研究是确定是否aleph 1-范畴是绝对的类的原子模型。PI还建议使用以前的结果相互代数结构的研究弱最小结构的扩展，保持stability.This奖项反映了NSF的法定使命，并已被认为是值得通过使用基金会的智力价值和更广泛的影响审查标准进行评估的支持。